The point of the choke balun (aka current mode balun) is it forces a balanced current condition on the balanced feed line. IOW, if you do indeed have common mode issues and not other reasons to explain the higher noise on your Doublet and you do tame that noise by adding more choking at the junction between the tuner and the balanced feed line. Then not only will noise ingress be reduced but you'll equally reduce any signal radiation from the feed line when you transmit. Noise ingress due to feed line imbalance and feed line transmission due to imbalance are flip sides of the same coin. Basically common mode currents on balanced feed line equal the residual when you take a vector sum of the currents on each branch of the line. If the currents are equal in magnitude and opposite in phase on the two balanced feed line conductors then their vector sum would be zero and there would be no common mode current. That's the completely balanced condition. When you have imbalance the two don't completely cancel and that residual when you sum the two currents IS the common mode current. If you introduce enough common mode impedance to successfully choke that current (or reduce it to negligible levels) you force the system back into balance. If the residual is forced close to zero then the two conductors have to be forced back into a balanced (or very nearly balanced with a negligible remainder) state to satisfy basic circuit principles like Kirchoff's current law or stated even more basically to satisfy conservation of charge. Bottom line, sufficient common mode rejection via choking improves both the receive and transmit performance of the feed line from the perspective of common mode issues.